intersect each other in A, the length of the curve AMO is equal to the radius of curvature PO.

Let the length of the arc AMO=s', then, (by Prop. 45. Cor. 1.), we have

ds' dy' dx'2
= +
dx2 dx2 dx;

but, in the former corollary, we found that

r.

and = - p. {3p - r . ( 1 + p')};

dx'

dx

dx'

(by Prop. 49. Cor. 1.);

hence, R and (s') are two functions of (x), which have the same differential coefficient; and, therefore, the difference between them is constant; but, when P coincides with A, O also coincides with A; and, therefore, their difference then = 0; consequently, their difference always = 0; and, therefore, we have R always s', or the arc AMO = PO, the radius of

curvature.

=

COR. 3. Hence, it is manifest, that if a thread be wound upon the curve OMA, then, by unwinding it, and keeping it stretched, the end P may be made to trace out the curve AP. In consequence of this property, the curve AMO is said to be the evolute of the curve AP, and the curve AP the involute of AMO.

SECT. VIII.

CURVATURE OF SPIRALS.

PROP. LXI.

There cannot be two unequal circles, which, when considered as spirals, have, at any point, their radii vectores, and the first and second differential coefficients of their radii vectores, respectively equal.

LET AP be a cir

cle whose centre is O, considered as a spiral, referred to the point S as its pole, and SL as its initial line; let the radius vector SP = p, the angle LSP = 0,

SO=c, OP=r, and SY the perpendicular upon the

tangent at P=p; then we have

SO2 = SP2 + PO3 - 2 SP. PO.cos SPO,

or c2 = p2+r2 = 2rp.cos PSY

= p2+r2 - 2r. p.

Hence, differentiating, we have

Similarly, if there be any other circle, considered as a spiral, and referred to the same pole and initial line, whose radius vector, radius, and perpendicular upon the tangent, are p', r' and p' respectively, it may be shown, that

we have, from the equations (4) and (C), p=p'; and, consequently, from the equations (B) and (D),

that is, the radii of the two circles are equal; and, consequently, the circles themselves cannot be unequal.

PROP. LXIÍ.

If a curve SPQ and a circle Pq', considered as spirals, referred to the same pole S, and the same initial line SL, have at any point P, a common tangent YPR; then according as at that point, the second differential coefficient of the radius vector of the circle, is less or greater than that of the radius vector of the curve, the circle will lie nearer to, or further from, the pole, than the curve.

cutting the circle in (g), the curve in Q, and the tangent in R, and let PSR=1. Then we have

(by Prop. 50. Cor. 10.); hence, SQ-Sq is equal

sufficiently small, this series is positive; and, therefore, SQ is greater than Sq; consequently, (q') is nearer to S than Q; and, therefore, the circular arc Pq is nearer to the pole than PQ, the arc of the In the same manner it may be shown, that a small arc of the circle Pq', measured on the other side of P, lies nearer to the pole, than the corre

curve.

COR. 1. If the spiral be concave, it falls between

d2p

do

be less than

the circle falls between the spiral and the pole; and, consequently, the circle falls within the spiral; and, therefore, its curvature is not less than that of the curve.

vature of the circle is not greater than that of the

curve.

COR. 3. If the spiral be convex, and

dεp than d023

the circle will fall without the spiral, (that

is, between the spiral and the tangent); and, therefore, its curvature will not be greater than that of the

curve.